1. Field of the Invention
The present invention relates to a method and apparatus for phase retrieval in wave phenomenon and may be used to provide multi-dimensional ultrafast laser diagnostics, among other applications.
2. Background
Interference phenomena are produced through the interaction of at least two spatially distinguishable waves. Sometimes diffraction and interference are not clearly distinguished. Interference occurs when two or more wavefronts interact while diffraction occurs naturally when a single wave is limited in some way. The nature of an interference pattern depends on several factors including the amplitudes and phases of the incoming waves. If the incoming waves are in phase, then the amplitude of the waves may add (constructive interference) whereas, if the phases of the incoming waves are separated by 180 degrees, then cancellation may result (destructive interference). For example, if light from a single laser is split to produce two separate beams and directed along two different paths, then it is likely that the beams will no longer be in phase or phase coherent. If the two beams are then recombined, the result will be an interference pattern. In some instances, it is helpful to detect the interference pattern as an image in a plane or on a planar surface. The interference pattern can also be captured as intensity (absolute value of amplitude squared) versus time in a two-dimensional detector. The intensity versus time patterns resulting from the interference between the two beams can be reconstructed if the system knows the individual frequency components, i.e., amplitudes and phases of the interference patterns.
Phase differences in the interference waves can be represented as shifts in origin. For example, the Fourier transform of a function f(x) (which converts a wave from amplitude as a function of time or space to amplitude as a function of frequency) is as follows:F(k)=∫f(x)exp(−ikx)dx  (1)where the limits on the integral are from x=−∞ to x=+∞. When the origin or phase is shifted, the Fourier transform is represented as,F1(k)=∫f(x−x0)exp(−ikx)dx  (2)or as,F1(k)=F(k)exp(−ikx0).  (3)Thus, the form of the Fourier transform differs only by the phase factor exp(−ikxo), remembering that the amplitudes |F1(k)| and |F(k)| and the intensity, the amplitudes squared, are equal. In many wave problems the function f(x) is complex, i.e., f(x)=Re[f(x)]+iIm[f(x)], and the transforms are of complex functions. A plane wave has a wavefront in a plane of constant phase normal to the direction of propagation. A plane wave may be written asE=E0exp(−iωt).  (4)In equation 4, the wave is represented as E, as a function of time, where the frequency characteristics are captured in the exponential term. In Equation 4, the amplitude is at a maximum at t=0.
If phase information for two interfering pulses is not known ahead of time, then the plot of intensity versus time of the interfering pulses does not uniquely define a set of frequencies and amplitudes for the pulses. There is not enough information in the intensity versus time plot alone to be able to reconstruct the phase of the original pulses. Determining the phase in these types of problems is referred to as phase retrieval. The phase retrieval problem is similar to solving a single equation for two unknowns—many solutions exist. Phase retrieval analysis often imposes a constraint, i.e., an additional equation, to overcome this problem. Typically the phase retrieval analysis selects a physically reasonable constraint that leads to a unique solution. The type of constraint depends on the application. In most crystallographic applications, the analysis typically applies a symmetry condition to the outer bounds of the measured region. Such constraints benefit from prior knowledge of the way atoms are arranged in a crystalline structure. Analysis of other applications requires that other constraints be found to facilitate phase retrieval.
Solution of phase retrieval problems starts with an initial estimate of the phase. Often the guess is not so critical and any reasonable starting point can be used to obtain a solution. Of course, because many of the phase retrieval techniques use iterative processes, a more reasonable guess typically results in fewer iterations in arriving at a unique solution.
One class of phase retrieval problems is known as reconstruction from multiple Fourier intensities (RMI). This problem involves the reconstruction of two functions from multiple Fourier intensities of the product based on relative displacements of the two functions. The solutions to this class of problems have significant applications to situations where the intensity of the Fourier transform of the product of two functions is recorded for multiple relative displacements of the functions including transmission microscopy (optical, electron, and x-ray) and ultrafast laser diagnostics. For example, in the field of ultrafast laser diagnostics, the solution to the RMI problem is known as a frequency-resolved optical gating (FROG) trace inversion, spectrogram inversion, or sonogram inversion and can be used to find the intensity and phase of an ultrashort laser pulse. In the FROG example (see FIG. 1), the system scans a gate pulse in time across a pulse to be measured. For each time delay, the spectrum of the pulse that results from a well-defined nonlinear interaction of these two pulses is recorded. The nonlinear interaction produces the product (multiplication) of the gate and pulse. The resulting spectrogram, or FROG trace, is a plot of intensity versus time and frequency of the pulse. It is only possible to obtain intensity information of the spectrum. Consequently, the key characteristics of the pulse, the intensity and phase, cannot be obtained directly from this plot. An iterative two-dimensional phase retrieval method can be used to find the phase in order to extract the functions, and hence, the pulse characteristics from the spectrogram. Current two-dimensional phase retrieval methods require a priori knowledge of the gate function or they are slow and cumbersome, requiring large amounts of computational power. There is a need for fast inversion methods for ultrashort pulse measurement devices, and the same method will be generally applicable to other fields. Indeed, even though inversion techniques exist for spectrograms and sonograms when the gate is known, there is still a need for fast, simple inversion methods for situations where the gate is known—especially for ultrafast laser diagnostics.
Ultrafast laser systems have a large number of applications in biochemistry, chemistry, physics, and electrical engineering. These systems generate laser pulses with durations of ten picoseconds or less and are used to explore kinetics in proteins, examine carrier relaxation in semiconductors, or image through turbid media. They are also used as an ultrafast probe in electronic circuits. By using ultrafast diagnostic systems, highly advanced semiconductors, electronic circuitry, and even biomedical products can be developed and tested for commercial applications. New applications requiring shaped ultrashort pulses in both intensity and phase such as coherent control of chemical reactions are being developed. The continued development of these applications will require fast, high quality and easy-to-use ultrafast laser pulse diagnostics.
FROG is an ultrafast laser diagnostic that is used to measure the intensity and phase of an ultrashort laser pulse. In a simple form, FROG produces a spectrogram of the pulse that is a three-dimensional plot of intensity versus frequency and time delay, showing the spectral components of time slices of the pulse. While the spectrogram of the pulse serves as an intuitive display of the pulse, it is difficult to obtain quantitative information about the pulse from the spectrogram, and subtleties in the pulse structure may go unnoticed without knowledge of the actual pulse. To obtain the original pulse from its spectrogram, the phase of the spectrogram must be determined requiring a two-dimensional phase retrieval computation. This mathematical step—which converts the measured spectrogram into two-dimensional plots of pulse duration and chirp—is the slowest step in existing FROG instrumentation. While it is possible to characterize an individual femtosecond pulse, the data analysis step might take sixteen-orders of magnitude longer than the duration of the pulse itself. A simple error analysis may take hours. More than just faster computers is needed to invert FROG traces in real time.
The development of techniques for ultrashort pulse measurement, that is, the profiling of the electric field envelope and the instantaneous frequency, has been difficult. Early methods yielded only the intensity autocorrelation of the pulse. Later developments, such as interferometric autocorrelation, achieved the indirect determination of various phase distortions common to ultrashort pulses, but complete intensity and phase information about the pulse remains difficult to obtain. Some work has been done to extract the time-dependent intensity I(t) and the phase φ(t) (or, essentially equivalent to the phase, the instantaneous frequency ω(t)), from these traces using iterative methods.
Fundamental inherent ambiguities, including the direction of time, remain. It is therefore not possible to determine, for example, the sign of the chirp, unless a second measurement is made after pulse propagation through a known dispersive medium. Other methods yield only I(t) or require a streak camera and hence lack sufficient temporal resolution. Still other methods have been developed to measure the phase ω(t) but do not yield the intensity. Simultaneous time and frequency information is required for retrieval of the full complex electric field.
Time-frequency measurements of ultrashort pulses were first completed by Treacy in 1971. The Treacy method disperses the input pulse in frequency, selects a portion of the frequency components to produce another pulse, then cross correlates the newly formed pulse with the original input pulse. By scanning the frequency filter over all of the frequencies contained in the original pulse, a three-dimensional plot of intensity versus frequency and time is produced which is commonly referred to as a sonogram. This method was refined by Chilla and Martinez with the development of frequency domain phase measurement or FDPM. Since the arrival time (i.e., the peak) of each frequency filtered pulse is given by the derivative of the phase (in the frequency domain) with respect to frequency, integration of arrival time of each pulse with respect to frequency gives the phase of the pulse in the frequency domain. Coupling this result with the spectrum of the pulse gives the Fourier transform of the complex electric field. The principal difficulty with this method is that if the peak of the arrival time of each frequency selected pulse does not produce a function, as is the case with self-phase modulated pulses, the group delay is not well defined and characterization of the pulse is not possible.
Spectrograms are close relatives to sonograms. Rather than displaying the time arrival of frequency filtered pulses, a spectrogram displays the frequency content of time slices of a pulse. FROG uses optical methods to obtain a spectrogram of the pulse to be measured to completely characterize ultrashort laser pulses. FROG can be used in either multi-shot or single shot geometries. A gate pulse, which can be virtually any duration, slices out portions of a probe pulse in the time domain using either an instantaneously responding nonlinear material or a nearly instantaneously responding medium. The sampled portion of the probe, or signal, is dispersed in a spectrometer. Like a sonogram, the resulting spectrogram contains all the intensity and phase information about the probe pulse.
Obtaining the spectrogram of a pulse is experimentally less complex than obtaining the sonogram; however, extracting the intensity and phase of a pulse from its spectrogram is mathematically more challenging. If an approach similar to that of Chilla and Martinez is used to invert a spectrogram, the phase in the time domain is obtained, but the complete complex electric field is not. Since only the magnitude of the spectrogram can be measured, finding the full intensity and phase of the input pulse requires determining the spectrogram's phase, placing the FROG inversion problem into the category of two-dimensional phase retrieval problems.
Mathematical Representation of an Optical Pulse
The discussion now reviews the mathematical representation of ultrashort optical pulses to provide background. A pulse's electric field, A(t), can be written:A(t)=Re[E(t)eiω0t]  (5)where ω0 is the carrier frequency and Re refers to the real part. A(t) can be used in this form for calculations, but it is generally easier to work with a different representation that removes the rapidly varying ω0 part, eiω0t, and uses as a representation a slowly varying envelope together with a phase term that contains only the frequency variations. This representation, which does not include the rapidly varying carrier frequency, isE(t)=[I(t)]1/2eiφ(t)  (6)where I(t) and φ(t) are the time-dependent intensity and phase of the pulse. (E(t) is complex.) The frequency variation, Ω(t), is the derivative of φ(t) with respect to time:Ω(t)=−dφ(t)/dt  (7)
The pulse field can be written equally well in the frequency domain by taking the Fourier transform of equation 6:{tilde over (E)}(ω)=[Ĩ(ω)]1/2e−iφ(ω)  (8)where Ĩ(ω) is the spectrum of the pulse and φ(ω) is its phase in the frequency domain. The spectral phase contains time versus frequency information. That is, the derivative of the spectral phase with respect to frequency yields the time arrival of the frequency, or the group delay.
Obtaining the intensity and phase, I(t) and φ(t) (or Ĩ(ω) and φ(ω)) is called full characterization of the pulse. Common phase distortions include linear chirping, where the phase (either in the time domain or frequency domain) is parabolic. When the frequency increases with time, the pulse is said to have positive linear chirp; negative linear chirp is when the high frequencies lead the lower frequencies. Higher order chirps are common, but for these, differentiation between spectral and temporal chirp is required because spectral phase and temporal phase are not interchangeable.
Frequency-Resolved Optical Gating
Frequency-resolved optical gating (FROG) measures, as shown in FIG. 1, the spectrum of a particular temporal component of an optical pulse by spectrally resolving the signal pulse in an autocorrelation-type experiment using an instantaneously responding nonlinear medium. As shown in FIG. 1, FROG involves splitting a pulse and then overlapping the two resulting pulses in an instantaneously responding χ(3) or χ(2) medium. Any medium that provides an instantaneous nonlinear interaction may be used to implement FROG. Perhaps the most intuitive is a medium and configuration that provides polarization gating.
For a typical polarization-gating configuration, induced birefringence due to the electronic Kerr effect is the nonlinear-optical interaction. The “gate” pulse causes the χ(3) medium, which is placed between two crossed polarizers, to become slightly birefringent. The polarization of the “gated” probe pulse (which is cleaned up by passing through the first polarizer) is rotated slightly by the induced birefringence, allowing some of the “gated” pulse to leak through the second polarizer. This is referred to as the signal. Because most of the signal emanates from the region of temporal overlap between the gate pulse and the probe pulse, the signal pulse contains the frequencies of the “gated” probe pulse within this overlap region. The signal is then spectrally resolved, and the signal intensity is measured as a function of wavelength and delay time τ. The resulting trace of intensity versus delay and frequency is a spectrogram, a time- and frequency-resolved transform that intuitively displays the time-dependent spectral information of a waveform.
The resulting spectrogram can be expressed as:SE(ω,τ)=|∫−∞∞E(t)g(t−τ)e−iωtdt|2  (9)where E(t) is the measured pulse's electric field, g(t−τ) is the variable-delay gate pulse, and the subscript E on SE indicates the spectrogram's dependence on E(t). The gate pulse g(t) is usually somewhat shorter in length than the pulse to be measured, but not infinitely short. This is an important point: an infinitely short gate pulse yields only the intensity I(t) and conversely a CW gate yields only the spectrum I(ω). On the other hand, a finite-length gate pulse yields the spectrum of all of the finite pulse segments with duration equal to that of the gate. While the phase information remains lacking in each of these short-time spectra, having spectra of an infinitely large set of pulse segments compensates this deficiency. The spectrogram nearly uniquely determines both the intensity I(t) and phase φ(t) of the pulse, even if the gate pulse is longer than the pulse to be measured. If the gate is too long, sensitivity to noise and other practical problems arise.
In FROG, when using optically induced birefringence as the nonlinear effect, the signal pulse is given by:Esig(t,τ)∝E(t)|E(t−τ)|2  (10)So the measured signal intensity IFROG(ω,τ), after the spectrometer is:IFROG(ω,τ)=|∫−∞∞E(t)|E(t−τ)|2e−iωtdt|2  (11)The FROG trace is thus a spectrogram of the pulse E(t) although the gate, |E(t−τ)|2, is a function of the pulse itself.
FROG is not limited to the optical Kerr effect. Second harmonic generation (SHG) FROG can be constructed to analyze relatively weak pulses from oscillators and is typically more sensitive than polarization-gating FROG. For SHG FROG, the pulse is combined with a replica of itself in an SHG crystal as illustrated in FIG. 2.
To see that the FROG trace essentially uniquely determines E(t) for an arbitrary pulse, note that E(t) is easily obtained from Esig(t, τ). Equation (11) can then be written in terms of Esig(t, Ω), the Fourier transform of the signal field Esig(t, τ) with respect to delay variable τ. This gives the following, apparently more complex, expression:IFROG(ω,τ)=|∫−∞∞Esig(t,Ω)e−iωt−iΩτ)dtdΩ|2  (12)Equation (12) indicates that the problem of inverting the FROG trace IFROG(ω,τ) to find the desired quantity Esig(t, τ) is that of inverting the squared magnitude of the two-dimensional Fourier transform of Esig(t, τ), which is the two-dimensional phase-retrieval problem discussed above. At first glance, this problem appears unsolvable; after all, much information is lost when the magnitude is taken. The one dimensional phase retrieval problem is known to be unsolvable (for example, infinitely many pulse fields give rise to the same spectrum). Intuition fails badly in this case, however. Two- and higher-dimension phase retrieval processes essentially always yield unique results.FROG Inversion
An iterative two-dimensional phase retrieval process is used to extract the pulse information from the measured FROG trace as illustrated generally in FIG. 3. This phase retrieval process converges to a pulse that minimizes the difference between the measured and the calculated FROG trace. Application of this phase retrieval process to FROG has been problematic in the past. Some recent applications use a generalized projections algorithm, which converges quickly, along with faster computers, to track pulse changes at rates of 20 Hz or greater, making FROG a real-time pulse measurement technique. Indeed, programs for analyzing FROG traces in this manner are commercially available.
The original FROG inversion process, using what is commonly referred to as the vanilla algorithm, is simple and iterates quickly. On the other hand, the process tends to stagnate and give erroneous results, especially for geometries that use a complex gate function such as SHG or self-diffraction. Improved strategies using different algorithms, including brute force minimization, were developed to avoid stagnation, at the expense of both iteration speed and convergence time. Later a numerical technique called generalized projections brought a significant advance in both speed and stability. The generalized projections technique proceeds after an iteration by constructing a projection that minimizes the error (distance) between the FROG electric field, Esig(t, τ), obtained immediately after the application of the intensity constraint, and the FROG electric field calculated from the mathematical form constraint. The projection constructed in this manner is used as the starting point of the next iteration.
The first implementations of the generalized projections technique used a standard minimization procedure to find the electric field for the next iteration (which can still be slow). For the most common FROG geometries, PG and SHG, there are substantial advantages to a different strategy that directly determines the starting point for the next iteration. This strategy, called principal components generalized projections (PCGP), converts the generalized projections technique into an eigenvector problem. The PCGP technique has achieved pulse characterization rates of 20 Hz.
The goal of phase retrieval is to find the E(t) that satisfies two constraints. The first constraint is the FROG trace itself which is the magnitude squared of the one dimensional Fourier transform of Esig(t,τ):IFROG(ω,τ)=|∫−∞∞Esig(t,τ)e−iωtdt|2  (13)The other constraint is the mathematical form of the signal field, Esig(t,τ), for the nonlinear interaction used. The mathematical forms for a variety of FROG beam geometries are:
                                          E            sig                    ⁡                      (                          t              ,              τ                        )                          ∝                  {                                                                                          E                    ⁡                                          (                      t                      )                                                        ⁢                                                                                                          E                        ⁡                                                  (                                                      t                            -                            τ                                                    )                                                                                                            2                                                                                                PG                  ⁢                                                                          ⁢                  FROG                                                                                                                                                E                      ⁡                                              (                        t                        )                                                              2                                    ⁢                                                            E                      *                                        ⁡                                          (                                              t                        -                        τ                                            )                                                                                                                    SD                  ⁢                                                                          ⁢                  FROG                                                                                                                          E                    ⁡                                          (                      t                      )                                                        ⁢                                      E                    ⁡                                          (                                              t                        -                        τ                                            )                                                                                                                    SHG                  ⁢                                                                          ⁢                  FROG                                                                                                                                                E                      ⁡                                              (                        t                        )                                                              2                                    ⁢                                      E                    ⁡                                          (                                              t                        -                        τ                                            )                                                                                                                    THG                  ⁢                                                                          ⁢                  FROG                                                              }                                    (        14        )            where PG is polarization gate, SD is self-diffraction, SHG is second harmonic generation and THG is third harmonic generation FROG.
All FROG implementations work by iterating between two different data sets: the set of all signal fields that satisfy the data constraint, IFROG(ω, τ), and the set of all signal fields that satisfy equation 14. The difference between FROG implementations is how the iteration between the two sets is completed. In the case of generalized projections, the E(t)'s are chosen such that the distance between the E(t) on the magnitude set and the E(t) on the mathematical form set is minimized. This is accomplished by minimizing the following equation:
                    Z        =                              ∑                          i              ,                              j                =                1                                      N                    ⁢                                          ⁢                                                                                                        E                                          sig                      ⁢                                                                                          ⁢                                              (                        DC                        )                                                                                    (                      k                      )                                                        ⁡                                      (                                                                  t                        i                                            ,                                              τ                        j                                                              )                                                  -                                                      E                                          sig                      ⁢                                                                                          ⁢                                              (                        MF                        )                                                                                    (                                              k                        +                        1                                            )                                                        ⁡                                      (                                                                  t                        i                                            ,                                              τ                        j                                                              )                                                                                      2                                              (        15        )            where Esig(DC)(k)(ti,τj) is the signal field generated by the data constraint, and Esig(MF)(k+1)(ti,τj) is the signal field produced from one of the beam geometry equations 14. For the normal generalized projections technique, the minimization proceeds using a standard steepest decent algorithm; the derivative of Z with respect to the signal field is computed to determine the direction of the minimum. The computation of the derivatives is tedious; the derivatives are tabulated in, for example, Trebino, et al., Rev. Sci Instrum., 68, p. 3277 (1997).
An alternative to the minimization of equation 15 is principal components generalized projections (PCGP). PCGP computes the starting point of the next iteration through an eigenvector problem, reducing the computation for the next iteration to simple matrix-vector multiplies. PCGP works for both the PG and SHG beam geometries, is simple to program and is fast, as described in D. J. Kane, IEEE J. Quant. Elec., (1999), and was used in the initial implementations of real-time FROG.
It is instructive to examine the PCGP strategy in detail to establish simplifying nomenclature. The PCGP strategy is based on the idea that FROG traces can be constructed from the outer product of two discrete vector pairs. Two vectors of length N are used to represent the probe and the gate fields:EProbe=[E1,E2,E3,E4, . . . ,EN]EGate=[G1,G2,G3,G4, . . . ,GN]  (16)The outer product of EProbe and EGate is:
                              [                                                                                          E                    1                                    ⁢                                      G                    1                                                                                                                    E                    1                                    ⁢                                      G                    2                                                                                                                    E                    1                                    ⁢                                      G                    3                                                                                                                    E                    1                                    ⁢                                      G                    4                                                                              …                                                                                  E                    1                                    ⁢                                      G                    N                                                                                                                                            E                    2                                    ⁢                                      G                    1                                                                                                                    E                    2                                    ⁢                                      G                    2                                                                                                                    E                    2                                    ⁢                                      G                    3                                                                                                                    E                    2                                    ⁢                                      G                    4                                                                              …                                                                                  E                    2                                    ⁢                                      G                    N                                                                                                                                            E                    3                                    ⁢                                      G                    1                                                                                                                    E                    3                                    ⁢                                      G                    2                                                                                                                    E                    3                                    ⁢                                      G                    3                                                                                                                    E                    3                                    ⁢                                      G                    4                                                                              …                                                                                  E                    3                                    ⁢                                      G                    N                                                                                                                                            E                    4                                    ⁢                                      G                    1                                                                                                                    E                    4                                    ⁢                                      G                    2                                                                                                                    E                    4                                    ⁢                                      G                    3                                                                                                                    E                    4                                    ⁢                                      G                    4                                                                              …                                                                                  E                    4                                    ⁢                                      G                    N                                                                                                      ⋮                                            ⋮                                            ⋮                                            ⋮                                            ⋮                                            ⋮                                                                                                          E                    N                                    ⁢                                      G                    1                                                                                                                    E                    N                                    ⁢                                      G                    2                                                                                                                    E                    N                                    ⁢                                      G                    3                                                                                                                    E                    N                                    ⁢                                      G                    4                                                                              …                                                                                  E                    N                                    ⁢                                      G                    N                                                                                ]                .                            (        17        )            This is referred to as the outer product form.
The rows of the outer product form are manipulated to generate an equivalent matrix that gives a time domain representation of a FROG trace. By leaving the first row unshifted and by shifting subsequent rows to the left, the following matrix results:
                    [                                                                                                                                     E                      1                                        ⁢                                          G                      1                                                                                                                                  E                      1                                        ⁢                                          G                      2                                                                                                                                  E                      1                                        ⁢                                          G                      3                                                                                        …                                                                                            E                      1                                        ⁢                                          G                                              N                        -                        2                                                                                                                                                        E                      1                                        ⁢                                          G                                              N                        -                        1                                                                                                                                                        E                      1                                        ⁢                                          G                      N                                                                                                                                                              E                      2                                        ⁢                                          G                      2                                                                                                                                  E                      2                                        ⁢                                          G                      3                                                                                                                                  E                      2                                        ⁢                                          G                      4                                                                                        …                                                                                            E                      2                                        ⁢                                          G                                              N                        -                        1                                                                                                                                                        E                      2                                        ⁢                                          G                      N                                                                                                                                  E                      2                                        ⁢                                          G                      1                                                                                                                                                                                                      ⁢                                                                  E                        3                                            ⁢                                              G                        3                                                                                                                                                        E                      3                                        ⁢                                          G                      4                                                                                                                                  E                      3                                        ⁢                                          G                      5                                                                                        …                                                                                            E                      3                                        ⁢                                          G                      N                                                                                                                                  E                      3                                        ⁢                                          G                      1                                                                                                                                  E                      3                                        ⁢                                          G                      2                                                                                                                                                              E                      4                                        ⁢                                          G                      4                                                                                                                                  E                      4                                        ⁢                                          G                      5                                                                                                                                  E                      4                                        ⁢                                          G                      6                                                                                        …                                                                                            E                      4                                        ⁢                                          G                      1                                                                                                                                  E                      4                                        ⁢                                          G                      2                                                                                                                                  E                      4                                        ⁢                                          G                      3                                                                                                                    ⋮                                                  ⋮                                                  ⋮                                                  ⋮                                                  ⋮                                                  ⋮                                                  ⋮                                                                                                                        E                      N                                        ⁢                                          G                      N                                                                                                                                  E                      N                                        ⁢                                          G                      1                                                                                                                                  E                      N                                        ⁢                                          G                      2                                                                                        …                                                                                            E                      N                                        ⁢                                          G                                              N                        -                        3                                                                                                                                                        E                      N                                        ⁢                                          G                                              N                        -                        2                                                                                                                                                        E                      N                                        ⁢                                          G                                              N                        -                        1                                                                                                                  ]                    .                                          ⁢                                                                                                                              ⁢                                      τ                    =                    0                                                                                                τ                  =                                      -                    1                                                                                                τ                  =                                      -                    2                                                                              …                                                              τ                  =                                      +                    3                                                                                                τ                  =                                      +                    2                                                                                                τ                  =                                      +                    1                                                                                                          (        18        )            The τ=0 column is the first column, where τ is the time delay in resolution element number (column number), which is just the probe multiplied by the gate with no time shift between them. The next column is the τ=−1 column where the gate is delayed relative to the probe by one resolution element. After some column manipulation the most negative τ is on the left and the most positive on the right; this time domain FROG trace is the discrete version of the product EProbe(t)EGate(t−τ). The columns are constant in τ (relative delay time) while the rows are constant in t (time). To obtain the FROG trace, the PCGP process obtains the Fourier transform of the product EProbe(t)EGate(t−τ) with respect to t. Each column of the matrix shown in equation 18 is Fourier transformed using a fast Fourier transform (FFT) or other suitable method. The final step of squaring the magnitude of the complex result produces the FROG trace.
Any invertible transform may be used in the PCGP process, but the most common invertible transform used is the Fourier transform. The PCGP process is also not limited to squaring of the magnitude of the complex result, or obtaining the magnitude of the complex result.
While there are an infinite number of complex images that have the same magnitude as the FROG trace to invert, there is only one image that is formed by the outer product of a single pair of vectors that has the same magnitude as the FROG trace to invert. Like other two-dimensional phase retrieval methods, a PCGP system uses an iterative method (FIG. 3) to find the proper vector pair. The PCGP process makes an initial guess for the phase of the FROG trace, and the result is decomposed into outer products. The principal pair of vectors is kept and used to determine the next guess of the FROG trace phase.
To construct the initial guess for the phase, a PCGP strategy constructs a FROG trace using vector pairs, one complex (probe) and, in the case of polarization-gate FROG, one real (gate), that are random noise modulated by a broad Gaussian pulse. The iterative method is started by replacing the magnitude of the newly constructed FROG trace by the square root of the magnitude of the experimental FROG trace.
The PCGP system converts the FROG trace with the correct magnitude to the time-domain FROG trace using an inverse Fourier transform by column (see FIG. 3). Next, the system converts the time-domain FROG trace to the outer product form, equation 17. If the intensity and phase of the FROG trace are correct, this matrix, which we will call O, is a matrix of rank 1. That is, OOT and OTO, where the superscript T refers to a transpose, has one and only one non-zero eigenvalue, which is the singular value of O.
The problem of producing the next best guess is solved using an elegant numerical method in linear algebra called singular value decomposition (SVD). SVD decomposes a matrix into a superposition of outer products of vectors. This can be written as:O=U×W×VT  (19)where U and VT are orthogonal (unitary, if they are complex) square matrices and W is a square diagonal matrix. Since both U and V are orthogonal, the column vectors of these matrices are all orthogonal and form an orthonormal basis set that describes the range of matrix O. Thus, the matrix O, the outer product form, is decomposed into a superposition of outer products between “probe” vectors (columns of U) and “gate” vectors (rows of VT). The diagonal values in W (the only non-zero elements of W) determine the relative weights of each outer product and, therefore, how much each outer product contributes to matrix O.
The PCGP system constructs a new FROG trace from the probe vector and the magnitude of the gate vector (or a complex valued gate vector when appropriate) obtained from the SVD of the outer product form matrix. The system repeats the process (see FIG. 3) until the FROG trace error, εFT, reaches an acceptably low value minimum.
Using SVD has an additional advantage as well. SVD provides the best packing efficiency for a given image. In other words, the image produced from the product of the outer product of the vector pair with the largest weighting factor is the best rank 1 approximation of that image in the least squares sense. That is, it minimizes
      ɛ    2    =            ∑              i        ,                  j          =          1                    N        ⁢                  ⁢                                  E          Outer                      i            ,            j                          -                              E            Probe            i                    ⁢                      E            Gate            j                                    where EOuter is the outer product form of the FROG trace shown in equation 17, EProbe is the probe vector, EGate is the gate vector, and ε is the error. This shows that the probe and gate found using SVD in the method described above represents a projection found directly without minimization. Neither the set of functions with a given Fourier transform magnitude nor the set of all outer products is a convex set, however. Since one or more of the constraints applied is not a projection onto a convex set, this method is related to the technique known as generalized projections; hence, the following discussion refers to this method as a principal component generalized projections method, or PCGPM.
Implementations of the FROG inversion method using the singular value decomposition may be comparatively slow and consume time by providing information with potentially limited use. Since only the principal eigenvector pair (one left and one right) is required, these eigenvectors can be obtained by using the power method. When a matrix, O, is decomposed using SVD, the eigenvectors of OOT (probe) and OTO (gate) are found along with the eigenvalues of OOT (which are the same as OTO and are called the weighting factors). The principal vector pair is the pair with the largest weighting value or the dominant eigenvalue of OOT (or OTO).
Assume a matrix O with a dominant eigenvector and xo is an arbitrary vector. If the vector is multiplied by the matrix O raised to a large power, the result is the dominant eigenvector multiplied by a constant with a negligible error. This works because the action of multiplying a vector by a matrix maps the vector onto a superposition of eigenvectors of the matrix. For example, suppose the vector xo is mapped onto a superposition of eigenvectors by matrix O. Then:(O)px0=k1λ1p−1v1+k2λ2p−1v2+ . . . +knλnp−1vn  (20)where v1 . . . vn are the eigenvectors of matrix O and k1 . . . kn are constants. It follows that:Ox0=k1v1+k2v2+ . . . +knvn  (21)where λ1 . . . λn are the eigenvalues of matrix O. The first multiplication of xo by matrix O maps xo onto a superposition of eigenvectors multiplied by constants k1 . . . kn. Each subsequent multiplication multiplies the eigenvectors by their respective eigenvalue. If p is large enough and if v1 is the dominant eigenvector, then λ1p−1, λ2p−1 . . . λnp−1. The result will appear to be the dominant, or principal, eigenvector multiplied by a constant.
If the PCGP strategy required a large number of matrix multiplications to find the eigenvector, then little would be gained by using this method over singular value decomposition. However, the PCGP strategy works quite well by replacing the SVD step with simple matrix vector multiplies:PROBEi=OOT(PROBEi−1)GATEi=OTO(GATEi−1)  (22)where i is the iteration number and O is the outer product form of the time domain FROG trace as described above. Thus, OOT maps the previous guess for the probe onto the next guess for the probe and OTO maps the previous guess for the gate onto the next guess for the gate. By replacing the SVD step by these mapping functions, the speed per iteration of the method is significantly increased while the number of iterations required for convergence is not increased. Further, the robust nature of the PCGP method is not compromised.Cross-Correlation FROG (X-FROG)
Cross-correlation FROG (X-FROG) uses a spectrally resolved cross-correlation between a known ultrafast laser pulse and an unknown ultrafast laser pulse to characterize the unknown pulse. There are several advantages to using a known pulse to characterize an unknown pulse. Wider wavelength ranges of ultrafast laser pulses can be measured; mid-IR pulses can be mixed with pulses to shift the measurement to inexpensive detectors. Complex, wide bandwidth ultrafast laser pulses, such as continua, can be characterized. Weak pulses can be measured by being cross-correlated with high power pulses. Like FROG, the extraction of the unknown pulse characteristics in X-FROG is usually accomplished using two-dimensional phase retrieval.
In the case of X-FROG, a known reference pulse or gate function is used to interrogate an unknown pulse in a cross-correlation configuration. At each time delay between the gate and the unknown pulse, the spectrum resulting from the nonlinear interaction of the pulse and gate, rather than the intensity, is measured, which forms a true spectrogram of the unknown pulse where the gate is the gating function for the spectrogram. Since the reference pulse (gate) is known a priori, and is cross-correlated with the unknown pulse to create the X-FROG trace, extraction of the unknown pulse is easier. In the case of the standard generalized projections algorithm, where a function is minimized, the known reference pulse (gate) is just part of the minimized equation.
Applying the PCGP strategy to X-FROG it is not as straight forward because the PCGP algorithm is naturally a blind algorithm—the pulse and the gate are found independently and no a priori information about the gate is used. While blind deconvolution may be used to find both the pulse and gate, this process is prone to error especially in the case of similar pulse shapes and durations; therefore, it would be much better to use the information about the gate (known reference pulse) in a PCGP retrieval. Unfortunately, using equation 22 with a known gate (or pulse) usually causes severe stagnation and so does not typically produce an algorithm that works well. Indeed, there has been a need for years for a strategy to use a known gate in the PCGP strategy to provide an X-FROG system that does not stagnate.
One method that was attempted by Dorrer and Kang, IEEE Photonics Technology Letters (2004), used a blind-PCGP algorithm for three iterations and then applied the known gate. Unfortunately, this is arbitrary and the number of iterations can be higher depending on the shape of the pulse being measured and the Dorrer and Kang strategy is prone to stagnation. The strategy is also not general purpose enough to be used in a product.